Extremals of a quadratic cost optimal problem on the real projective line

Let Σ be a bilinear control system on R² whose matrices generate the Lie algebra sl(2) of the Lie group Sl(2) : the group of order two real matrices with determinant 1. In this work we focus on the extremals of a quadratic cost optimal problem for the angle system PΣ defined by the projection of Σ onto the real projective line P¹. It has been proved in [2] that through the Cartan-Killing form the cotangent bundle of P¹ can be identified with a cone C in sl(2). Via the Pontryagin Maximum Principle, we explicitly show the extremals by using the mentioned identification and the special form of the trajectories associated with the lifting of vector fields on PΣ. We analyze both: the controllable case and when the system bf PΣ give rise to control sets. Some examples are shown.